Method for Faster Secure Multiparty Inner Product with SPDZ

ABSTRACT

A method for implementing a secure multiparty inner product computation between two parties using an SPDZ protocol involves having a first party and a second party compute, for i=k, a vector (I)=(II) based on a vector (x={ 1 , . . . , x N }), and a vector (w={W 1 , W N }), respectively, where (I)=(X 2i− X 2i ) (III)=W 2i−1 W 2i , N is the total number of elements in the vectors k=N/2. The vectors (I), and (III) are securely shared between the parties. The parties then jointly compute SPDZ protocol Add([w 2i ], [x 2i−1 ]) and Add([w 2i ], [x 2i−1 ]) to determine shares [w 2i−1 +x 2i ] and [w 2i +x 2i−1 ] respectively, and then compute, for i= 1, . . . ,  k, inner product shares [d i ] by performing SPDZ protocol Mult([w 2i31 1 +x 2i ], [w 2i +x 2i−1 ]). SPDZ protocol ([Add d 1 ],. . . , [d k ], −(IV), . . . , −(V), −(VI), −, (VII)) is

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application Ser. No. 62/598,666 enitled “ METHOD FOR FASTER SECURE MULTIPARTY INNER PRODUCT WITH SPDZ” by Zheng et al., filed Dec. 14, 2017, the disclosure of which is hereby incorporated herein by reference in its entirety.

TECHNICAL FIELD

This disclosure relates generally to multiparty computation (MPC) protocols, and, in particular, to the SPDZ (pronounced “speedz”) MPC protocol.

BACKGROUND

In a secure multiparty computation (MPC) protocol, a number of parties jointly compute a function on their private inputs and learn only the output of the function. The SPDZ protocol is one of the most practical and actively secure MPC protocols and achieves the security against any number of malicious corrupted parties. To be specific, the SPDZ protocol is working in the preprocessing model: Without knowing any future inputs, the offline phase executes the time-consuming tasks to generate tuples; and the online phase executes the specific functions with the private inputs, as well as leveraging those tuples from the offline phase. In particular, SPDZ is a general-purpose MPC protocol allowing joint computation of arithmetic circuits.

The secure inner product is a widely used building block in privacy-preserving data mining. As is well known to the art, the inner product operation (also referred to as a “dot product”) multiplies the corresponding pairs of elements from both vectors together to produce individual products and sums all of the products to produce a final scalar sum value. For example, if w=(1, 2, 3) and x=(4, 5, 6), then the inner product is: (1·4)+(2·5)+(3·6)=32. The inner product operation is used in a wide range of practical computations, and to cite one non-limiting example, neural networks make heavy use of inner product operations where the vector w represents a vector of weights used in the neural network and the vector x represents input values, where part of the operation of a neuron is to generate a sum of each weights multiplied by a corresponding input value, which is the inner product operation described above. The SPDZ protocol enables two or more parties that each has access to only part of the input data to generate the inner product value using secure multiparty computations. For example, in one configuration one computing device has access to the weight vector w in the neural network and another computing device has access to an input vector x, which may contain information about an input image or other input data that is privacy sensitive. The SPDZ protocol enables the computation of the final output of the inner product without revealing the input vector x to the party that stores the weight values w and without revealing the weight values w to the party that stores the input vector x. The SPDZ protocol enables this operation between groups of at least two parties, although in some configurations the vectors w and x are further subdivided amongst a larger group of parties. While not described in greater detail herein, the SPDZ protocol also provides some ability for parties in a group to determine if a subset of the parties has manipulated shared data to generate an incorrect final result, which provides some capability for the group to determine if the final computed result has been calculated accurately or should be discarded.

For two N-dimensional vectors x=(x₁, x₂, . . . , x_(N)) and w=(w₁, w₂, . . . , w_(N)), the inner product of them is the sum Σ_(i=1) ^(N)x_(i)w_(i). The straightforward implementation of secure inner product x·w with SPDZ involves N pairwise multiplications followed by N−1 additions in the online phase. Finding ways to reduce the computational requirements for secure inner products can improve the performance for both online and offline phases of SPDZ.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts the steps of the SPDZ Protocol Share(ct_(s), P₁, . . . , P_(n)) where a_(s)=Enc_(pk)(s) is publicly known and P₁, . . . , P_(n) are n parties.

FIG. 2 depicts the steps of the SPDZ Protocol Add([x¹], [x²], . . . , [x^(l)], where [x^(j)] is the share value x^(j), 1≤j≤1.

FIG. 3 depicts the step of the SPDZ Protocol Mult([x], [y]), where [x] is the share of value x and [y] is the share of value y.

FIG. 4 is a schematic diagram of a system that is configured to perform secure multiparty computation of inner products with improved efficiency.

FIG. 5 depicts the algorithm for a protocol ReShare for fault tolerant SPDZ in accordance with the present disclosure.

FIG. 6 depicts the algorithm for a protocol ReCombine for fault tolerant SPDZ.

FIG. 7 depicts a table of the intermediate results after a party redistributes its holding secret share to other parties.

FIG. 8 depicts a table showing the final results after each party redistributes their secret shares to the other parties.

FIG. 9 depicts a table showing the final result after executing protocol ReShare.

FIG. 10 depicts the algorithm for the secret share redistribution phase which is based on the protocol ReShare.

FIG. 11 depicts the algorithm for the secret share redistribution phase which is based on the protocol ReCombine.

FIG. 12 schematically shows the SPDZ Protocol architecture versus the proposed Fault Tolerant SPDZ Protocol architecture.

FIG. 13 depicts a table showing a comparison of the SPDZ Protocol and the proposed Fault Tolerant SPDZ Protocol.

DETAILED DESCRIPTION

For the purposes of promoting an understanding of the principles of the disclosure, reference will now be made to the embodiments illustrated in the drawings and described in the following written specification. It is understood that no limitation to the scope of the disclosure is thereby intended. It is further understood that the present disclosure includes any alterations and modifications to the illustrated embodiments and includes further applications of the principles of the disclosure as would normally occur to a person of ordinary skill in the art to which this disclosure pertains.

This disclosure is directed to an efficient method for computing multiparty inner product with SPDZ. At the expense of sharing additional precomputed data among parties, the approach according to this disclosure is able to reduce the number of multiplications as well as required Beaver triples by half for the online phase of the SPDZ protocol, thereby improving the performance significantly for both online and offline phases when computing multiparty inner product with SPDZ.

As used herein, the term “Beaver triple” refers to a set of three values (a, b, c) in which a and b are selected uniformly at random and c is equal to the product of a multiplied by b: c=ab. All three numbers in the Beaver triple are members of a finite field F_(p) where p is a modulo value. In some illustrative embodiments where p=2^(n)−1, the finite field F_(p) includes all integers in a range of [−2^(n/2), 2^(n/2)−1] where n is 64, 80, or 128, although other finite fields can be used. As described in further detail below, in a modified SPDZ protocol the parties generate Beaver triples during an offline computation phase and distribute shares of the Beaver triples in a secure manner for later use as part of an online computation phase for multi-party computation of inner products.

As used herein, the terms “homomorphism” and “homomorphic” refer to a property of some cryptographic systems in which mathematical operations can be applied to encrypted values to produce an encrypted result that matches the result of performing the same operations to the plaintext values without revealing the contents of the encrypted values or the result to the computing device that performs the operation. For example, given two values a=1 and b=2 and their corresponding encrypted counterparts a′ and b′, in a cryptographic system that provides additive homomorphism the encrypted value c′=a′ +b′ stores an encrypted representation of the number “3” (1+2=3). The value c′ can be computed by a computer that cannot determine the plaintext values of a′ or b′ and, in some instances, the computer that produces c′ can decrypt c′ to retrieve the plaintext sum (3) without the ability to determine the original plaintext values of a or b. Subtractive homomorphism similarly applies to an operation that subtracts two encrypted values (e.g. a=2, b=1, c′=a′−b′, decryption (c′)=1), and in the embodiments described herein subtractive homomorphism is also equivalent to additive homomorphism in which one or more selected values are negated to perform an addition operation that is equivalent to subtraction. Similarly, a cryptographic system that provides multiplicative homomorphism the product of two encrypted values (a′ and b′) is another encrypted value (c′) that decrypts to the same value as the product of the two plaintext values (a and b) (e.g. a=2, b=3, c′=(a′b′), decryption (c′)=6). In the embodiments described herein, the

notation refers to an addition between two encrypted values in a cryptographic system that provides additive homomorphism, the

refers to a subtraction operation between two encrypted values in a cryptographic system that provides subtractive homomorphism, where subtraction can be considered as addition of a negated value, and the

notation similarly refers to a multiplication between two encrypted values in a cryptographic system that provides multiplicative homomorphism.

As used herein, the term A refers to a message authentication code (MAC) cryptographic key that is selected from the finite field F_(p) (λ∈F_(p)). Each of the parties P_(i), 1≤i≤n, that perform the inner product operation are provided with an individual share λ_(i) of the overall secret A where A is the sum of all the individual secrets (λ=λ₁+ . . . +λ_(n)). None of the parties has access to the full MAC cryptographic key λ. The MAC cryptographic key λ can generate an encrypted MAC value that corresponds to a message authentication code for a value x via multiplication λx, and as described below the MAC key λ can itself be encrypted using a public key to generate an encrypted ciphertext of the MAC key ct_(λ) that can be shared amongst multiple parties. The encrypted MAC value is used for verification of information received from other parties and to provide resistance against malicious parties that alter shared data during multiparty computation operations. Some embodiments that are described herein may omit the use of the MAC key λ in configurations of multiparty computation systems that preserve privacy of the origin input data but that trust the participants to provide accurate data outputs during multiparty computations.

As used herein, the term ε=(Enc, Dec) refers to an asymmetric public key encryption system that provides at least additive/subtractive and multiplicative homomorphism with threshold decryption. The term sk refers to the private key in ε that is used for decryption of ciphertext with the Dec function and the term pk refers to the public key in ε that is used for the encryption of plaintext to produce ciphertext using the Enc function. Each of the parties P_(i), 1≤i≤n that performs the multiparty inner product computation has access to the full public key pk and a portion of the private key sk_(i) where sk=sk₁+ . . . +sk_(n) where none of the parties has access to the full private key sk.

As used herein, the notation [ ] for a value (e.g. [x]) indicates a “share” of the value or where multiple parties each hold a share [x] of the overall value x, but having access to a share [x] does not reveal the actual value of x. In the embodiments described below, the value x is an encrypted ciphertext value, and the additive homomorphic properties of the cryptographic systems used herein enable the total value x to be divided into individual shares that have a total value of x. In the computation of inner products from two vectors that have multiple elements, each element of a vector represents a value that is shared amongst two or more nodes in a secure manner. Mathematically, [x]={(x₁,γ₁(x)), . . . , (x_(n), γ_(n)(x))} for a total of n parties p_(i), 1≤i≤n where each share [x_(i)] is a portion of the total [x] where x=x₁+x₂+x_(n), the term γ_(i)(x) for each party is another individual share of the value H_(x), which is the output of the cryptographic key λ multiplied by the value x(H_(x)=λx), and given the individual shares γ_(i)(x), the total is H_(x)=γ₁(x)+γ₂(x)+ . . . +γ_(n)(x). Each party stores a share value [x] as one of the tuples (x_(i), γ_(i)(x)), and does not reveal this tuple to other parties in the group directly.

The SPDZ protocol uses a two-phase approach that includes an “offline” phase of computations that multiple parties use for the secure exchange of randomly generated numbers known as Beaver triples with each other. The Beaver triples are not the actual data of interest to the nodes, but are used in the subsequent online phase to enable computation of shared results while preventing each party from identifying the private data from one or more additional parties. The offline phase introduces computations to compute and share the Beaver triples that includes encryption operations, and is generally considered to be the more computationally intensive phase of the SPDZ protocol. In a second “online” phase the parties use the Beaver triples and the actual data of interest to perform secure multiparty computations, including the inner product computation that is described above. While the SPDZ protocol is effective at performing inner product computations, the large number of multiplication operations that occur in the inner product computation during the online phase introduces additional communication operations that each requires the use of Beaver triples and produces computational overhead between the parties due to the underlying operations that SPDZ implements for multiparty multiplications.

Before the offline and online phases, cryptographic keys must be generated, including public cryptographic parameters, public key, private key shares (possessed by each party) and MAC key shares (possessed by each party) are generated. In particular, each party receives one of n MAC key shares where n is the total number of parties. Suppose that λ is the MAC key and λ₁+ . . . +λ_(n) are secret (MAC) shares, then party P_(i), 1≤i≤n, possesses the secret share λ_(i). Note that this step is executed only once unless the MAC key needs to be updated.

During the offline phase of the SPDZ protocol, a sufficient number of multiplication triples, in particular Beaver multiplication triples, are generated which will allow the parties to compute products. The Beaver multiplication triples consist of three values ([a], [b], [c]) which are shares of a, b, c, respectively, where a, b, c∈F_(p) are random values and c=ab. These values can be generated by the crypto provider, and the resulting shares distributed to the parties before the online phase.

Given a ciphertext ct_(s)=Enc_(pk)(s) for some value s, SPDZ protocol Share can securely distribute the secret share of s, as shown in FIG. 1. Protocol Share (ct_(s), P₁, . . . , P_(n)) where ct_(s)=Enc_(pk)(s) is publicly known and P₁, . . . , P_(n) are n parties. Referring to FIG. 1, each party generates a value a value f_(i)∈F_(p) uniform at random, and encrypts it to a ciphertext ct_(f1)=Enc_(pk)(f_(i)) (Block 100). All parties then broadcast their ciphertexts so that each party can compute the ciphertext ct_(s+f) =ct_(s)

ct_(f1)

. . .

ct_(fn). (Block 102). All parties then cooperatively decrypt ct_(s+f) with their own secret share of the private key sk and obtain s+f, where f=Σ_(i=1) ^(n)f_(i) (Block 104). Party P₁ sets the secret share of s as s+f−f₁ and party P_(i)(i≠1) sets the secret share of s as −f_(i) (Block 106).

With the protocol Share, let ct_(λ)=Enc_(pk)(λ) be the ciphertext of the MAC key λ, then the process of generating the triple ([a], [b], [c]) is performed. The following steps may be used to generate [a]:

-   -   Party P_(i), chooses a value a_(i)∈F_(p) uniform at random, and         encrypts it to ct_(ai)=Enc_(pk) (a_(i))     -   One selected party computes ct_(a)=ct_(a) ₁         . . .         ct_(a) _(n) , as well as ct_(λa)=ct_(λ)         ct_(a).     -   All parties together execute Share(ct_(λa), P₁, . . . P_(n)),         such that party P_(i), 1≤i≤n, holds a secret share γ_(i) (a) of         λa. Therefore, Party i holds (a_(i), γ_(i)(a)) where a=a₁+ . . .         +a_(n) and λ×a=y₁(a)+ . . . +γ_(n) (a).

The same steps used to generate [a] are repeated to generate [b]. The value [c] is generated such that c=ab. Given ct_(a), and ct_(b) from steps 1 and 2, let ct_(c)=ct_(a)

ct_(b). The protocol Share(ct_(c), P₁, . . . P_(n)) is executed, so that Party i holds the secret share c_(i) of c. Given c+f when executing a protocol Share on ct_(c), a fresh ciphertext ct′_(c)=Enc_(pk)(c+f)

ct_(f1)

. . .

ct_(fn) can be computed where Enc_(pk)(c+f) is a fresh ciphertext by encrypting c+f, as well as ct_(λc)=ct_(λ)

ct_(c). All parties together execute Share (ct_(λc), P₁, . . . , P_(n)), so that Party P_(i), 1≤i≤n, holds a secret share γ_(i)(c) of λc. Eventually, Party i holds (c_(i), γ_(i)(c)), so that c=c₁+ . . . +c_(n) and λc=γ_(i)(c)+ . . . +γ_(n)(c).

As is known in the art, generating triples requires performing public key operations (i.e., homomorphic addition and multiplication over ciphertext) and therefore the offline phase is time-consuming for large number of triples. Note that a beaver multiplication triple is one kind of tuple; several other tuples, such as inverse tuples ([x], [1/x]), also exist in protocol SPDZ for different purposes.

Since any arithmetic circuit can be represented as a set of addition and multiplication gates, the online phase focuses on how to compute the addition and multiplication securely. A multiparty addition enables multiple parties to add different shares of I different values to generate a sum, and the multiparty addition also includes subtraction by negating the appropriate values to perform subtraction. An embodiment of the SPDZ Protocol Add is depicted in FIG. 2. In a simple example where I=2 the multiparty addition adds the shares of two numbers together, although a multiparty addition can add the shares of more than two numbers as well. The terms [x¹], [x²], . . . , [x^(l)] represent shares of I different values x¹, x², . . . , x^(l), respectively, that are being added in the multiparty addition process. In the context of this document the superscript notation does not indicate a mathematical exponent. As described above, each share includes a tuple (x_(i), y_(i)(x)). During multiparty addition, each party computes a value z, based on a sum of the x, terms from all of the shares: z_(i)=x_(i) ¹+x_(i) ²+ . . . +x_(i) ^(l) and a value γ_(i)(z) based on a sum of the y_(i)(x) terms from all of the shares: γ_(i)(z)=γ_(i)(x¹)+γ_(i)(x²)+ . . . +γ_(i)(x^(l)). These operations enable each of the parties P_(i), 1≤i≤n to have a share [z_(i)] of the total sum value z and the share γ_(i)(z) of the MAC value ct_(λ)z where each party holds the tuple (z_(i), γ_(i)(z)). As noted above, each party has a share [z_(i)] of the sum z, but each party does not have access to the actual sum value of z, which ensures that even in a two-party addition process that one party cannot determine the value held by the other party given the sum and the plaintext of one of the two values being added together.

In the multiparty addition described above, the computational complexity for each addition is

(l) for the I shares that are being added together, and the computations include comparatively simple addition operations without requiring the use of the public key pk for further encryption operations. Furthermore, after the shares [x¹], [x²], . . . , [x¹] have been distributed to the parties, there is no further need for communication between parties or the use of a Beaver triple to compute the share of the sum [z_(i)] for each party.

A multiparty multiplication enables the parties that have shares [x] and [y] of two values x and y. Protocol Mult in FIG. 3 shows how to securely perform multiplication. Unlike the multiparty addition that can add together shares from more than two values, the multiparty multiplication is applied to exactly two values. The multiparty multiplication process uses a Beaver triple ([a], [b], [c]) that has been shared in an offline phase that is described below. During the multiparty multiplication process, each party P_(i), 1≤i≤n uses the addition process that is described above to add shares of the input [x] and a negated Beaver triple share value [a]: Add ([x_(i)], −[a_(i)]). Similarly, each party uses the addition process that is described above to add shares of the input [y] and a negated Beaver triple share value [b]: Add([y_(i)], −[b_(i)]). Each party broadcasts the results of both additions described above to all of the other parties: ∈=x−a and ρ=y−b. As described above, because of additive homomorphism, the results ∈ and ρ reveal the total differences between the tuples of x and a and the tuples of y and b. The multiparty multiplication process continues using the c, value from each share [c] in the Beaver triple as each party generates two local values z, and γ_(i)(z):

$z_{i} = \left\{ {{\begin{matrix} {c_{i} + {\epsilon \; b_{i}} + {\rho \; a_{i}} + {\epsilon \; \rho}} & \left( {i = 1} \right) \\ {c_{i} + {\epsilon \; b_{i}} + {\rho \; a_{i}}} & \left( {i \neq 1} \right) \end{matrix}\mspace{14mu} {and}{\gamma_{i}(z)}} = {{\gamma_{i}(c)} + {\epsilon \; {\gamma_{i}(b)}} + {\rho {\gamma_{i}(a)}} + {\lambda_{i}\epsilon \; \rho}}} \right.$

The multiparty multiplication process enables each party to hold a share [z_(i)] of the product of the original values x and y (z=xy) where once again [z_(i)]=(z_(i), γ_(i)(z)). As with the multiparty addition process, even if there are only two parties that participate in a multiparty multiplication and each party is assumed to have access to one of the plaintext inputs x or y, the share [z_(i)] from the multiplication process prevents either party from identifying the value that is held by the other party because the share [z_(i)] does not reveal the true product value z. The multiparty multiplication process requires both communication between the parties to produce shared values ∈ and ρ. Additionally, the multiplication process consumes the shares ([a], [b], [c]) of one Beaver triple, where each Beaver triple is only used once.

FIG. 4 depicts a system 100 that includes a plurality of compute nodes 104A and 104B that are communicatively connected to each other via a data network 150. In the system 100, each of the nodes 104A and 104B is a party in the MPC processes described herein. Each node is a computing device that acts as a single party in the secure multiparty computations processes that are described herein, and the term “node” is used interchangeably with the term “party” in the embodiments described below. FIG. 4 depicts the node 104A in detail, and the node 104B is configured in a similar manner with different sets of stored data to implement a secure multiparty inner product computation using the embodiments described herein. While FIG. 4 depicts a system with two nodes 104A-104B for illustrative purposes, the embodiments described herein can be performed by groups of three or more nodes as well.

Referring to node 104A in more detail, the node includes a processor 108 that is operatively connected to a network interface device 112 and a memory 120. The processor 108 is typically a central processing unit (CPU) with one or more processing cores that execute stored program instructions 124 in the memory 120 to implement the embodiments described herein. However, other embodiments of the processor 108 use different processing elements instead of, or in addition to, a CPU including graphics processing units (GPUs), field programmable gate arrays (FPGAs), application specific integrated circuits (ASICs), digital signal processors (DSPs), and any other digital logic device that is configured to perform the operations described herein. In some embodiments, the processor 108 implements a hardware random number generator (RNG) or uses a software random number generator or pseudo-random number generator (PRNG) to generate the random values. In the description herein, any reference to the generation of a random value or use of a random value refers to the operation of an RNG or PRNG to generate a value in a uniformly random manner selected from a predetermined numeric range (e.g. the finite field F_(p)).

The network interface device 112 connects the node 104A to a data network 150, such as a local area network (LAN) or wide area network (WAN) to enable communication between the node 104A and the node 104B of FIG. 4, and to additional nodes that perform the secure multiparty inner product computation processes described herein. Non-limiting embodiments of the network interface device 112 include a wired network devices such as an Ethernet adapter or wireless network devices such as a wireless LAN or wireless WAN network adapter. In the system 100, all transmissions of data that are made through the network 150 are assumed to be observed by third party adversaries (not shown in FIG. 4) that can record all data transmitted from the nodes 104A and 104B, although the nodes 104A and 104B may communicate using a transport layer security (TLS) encrypted and authenticated channel or other equivalent communication channel that prevents eavesdroppers from observing or altering communications via the network 150. The embodiments described herein prevent an adversary that can observe communication between the nodes 104A and 1048 from determining the value of any private data in addition to preventing the node 104B from identifying the private plaintext data 128 of the node 104A and vice versa during multiparty inner product computations.

The memory 120 includes one or more volatile memory devices such as random access memory (RAM) and non-volatile memory devices such as magnetic disk or solid state memory devices that store the program instructions 124, private plaintext data 128, value share data 132, cryptographic key data 163, and the computed result of the inner produce 140. The private plaintext data 128 include all or a portion of one vector in two vectors that form the inputs to the inner product computation. The two vectors in the plaintext data 128 are referred to as x and w herein. The value share data 132 include the share that each node receives during the operation of the system 100 including shares of Beaver triple values that are generated during an offline phase and shares of values that are exchanged during the inner product calculation process. As described above, each share provides an encrypted version of a portion of each value to each node. The cryptographic key data 136 include a share of the encrypted MAC cryptographic key for the node 104A, the public key pk, and the share of the private key sk, that is shared with the node 104A. The computed result of the inner product 140 is the final output of the multiparty computation of the inner product value.

The node 104B includes a memory with similar data structures except that the node 104B stores a different set of private plaintext data 128 corresponding to the second vector used to generate the inner product, different sets of secret value shares 132, and different shares of the private key and MAC key in the cryptographic key data 136. As described below, both nodes 104A and 104B compute the same result of the inner product 140. In the illustrative embodiment of FIG. 4, the node 104A stores all of the plaintext elements of the vector x and the node 104B stores all of the plaintext elements of the vector w, but in other embodiments the plaintext data for either or both of the vectors is distributed to multiple nodes to enable a larger number of nodes to perform secure multiparty inner product calculations.

To be specific, in our setting, one party holds the input vector x={x₁, . . . , x_(N)} and the other party holds the weight vector w={w₁, . . . , w_(N)} and they wish to compute jointly the following equation (equation 1):

x ₁ w ₁ + . . . +x _(N) w _(N)=Σ_(i=1) ^(N) x _(i) w _(i)   (1).

Following is a description of the previously known protocol for secure inner product computation. Here we only focus on the online phase, and we assume that the offline phase has been set up with respect to the SPDZ protocol.

The previously known inner product is given by Protocol (IP) ([P₁, x₁, . . . , x_(N)], [P₂, w₁, . . . , W_(N)], ct_(λ)), where party P₁ holds a private vector x₁, . . . , x_(N), and party P₂ holds a private vector w₁, . . . , w_(N) and ct_(λ) is the ciphertext of the MAC key λ and only can be decrypted by the two parties together.

-   -   1: For i=1, . . . , N, Party 1 chooses a value x_(i2) uniform at         random, sets x_(i1)=x_(i)x_(i2), generates ct_(λx) _(i)         =Enc_(pk)(x_(i))         ct_(λ), sends x_(i2) to Party 2 and runs the protocol         ReShare(ct_(λx), P₁, P₂), so that x_(i) is securely shared         between party P₁ and party P₂, denoted by [x_(i)].     -   2: For i=1, . . . , N, Party 2 chooses a value withd it uniform         at random, sets w_(i1)=w_(i)−w_(i2), generates ct_(λw) _(i)         =Enc_(pk)(w_(i))         ct_(λ), sends w_(i1) to Party 1 and runs the protocol         ReShare(ct_(λx), P₁, P₂), so that x_(i) is securely shared         between party P₁ and party P₂, denoted by [w_(i)].     -   3: Party P₁ and Party P₂ run SPDZ protocol (mult[x_(i)],         [w_(i)]), 1≤i≤N, which is denoted as [d_(i)].     -   4: Party P₁ and Party P₂ then perform SPDZ protocol (Add[d_(i)],         . . . , [d_(N)]).

As noted above, one operation that occurs during the protocol is a secure resharing operation. A secure resharing operation, which is referred to as a “ReShare” or “ReSharing” operation or protocol, enables two or more nodes to generate shares of a value that is already known to all of the nodes (i.e. the full value is already “shared” by the nodes and this process “reshares” portions of the overall value) in which each node holds a share of the total value without knowing the share that is held by any other node. In the embodiments described herein, the reshare operation is defined as: Reshare (ct_(s), P₁, . . . P_(n)) where ct_(s) is an encrypted ciphertext (ct) of a secret value s and P₁ . . . P_(n) represent the n parties that participate in the resharing operation. The encrypted ciphertext value ct is known to all of the parties as a whole, and the resharing operation enables the parties to each receive a share of the original set of data without knowing the shares of the other nodes, even in a two-node system. The resharing operation begins with encryption of the secret value s using the public key pk to produce a ciphertext ct_(s)=Enc_(pk)(s), and the ciphertext ct_(s) is transmitted to all of the n parties in the group of parties that receive an encrypted share of the secret value s. Each party i in the n parties generates a uniformly random value f_(i)∈F_(p) and uses the public key pk to encrypt the random value to generate a ciphertext of the random value: ct_(fi)=Enc_(pk)(f_(i)). Each party transmits the respective ciphertext value ct_(fi) to all of the other parties via a data network, and each party calculates a ciphertext sum ct_(s+f) of the encrypted value of s and the individual encrypted values ct_(fi): ct_(s+f)=c_(s)

ct_(f) ₁

ct_(f) ₂

. . .

ct_(f) _(n) where once again the

operator indicates addition of ciphertexts in a cryptographic system that provides additive homomorphism. The parties then perform a cooperative decryption process using the individual portions of the private key ski to decrypt the value ct_(fi) to obtain the sum of s+f, where f is the sum of all the individual random values f_(i). The final shares amongst the n parties are [s₁]=s+f−f₁ for party 1 and [s_(i)]=−f_(i) for each of the other i parties.

The resharing operation described above is one method to distribute shares of a ciphertext value that is known to all of the parties in a group. Another operation is referred to as a “share distribution” operation that generate shares of a plaintext value that is known to one party in the group and that does not need to be encrypted before being shared between the nodes in the group while not disclosing the plaintext data to the other parties. As described below, this operation enables nodes in a system that perform multiparty inner product computations to distribute shares of individual elements in a vector and pair products based on the multiplication of two elements in a single vector where one node has access to the plaintext contents of the elements. Given a plaintext value that is referred to as x, the party that possesses the value x generates a random value using a uniform random distribution in F_(p) for each of the other parties in the group, subtracts all of the randomly generated values from the original value x to generate a share [x] that is used by the original party, and transmits the randomly generated values to one or more additional parties where the random values serve as the shares for [x] used by the other parties. For example, in a group with two parties a first party that holds x_(i) in memory, where i indicates the index of the value in a vector, and generates a random value that is labeled x_(i2) where the 2 is an index label that indicates another node that receives the random value. The first party generates a share of x_(i) that is labeled [x_(i1)]=x_(i)−x_(i2) based on the difference between x_(i) and x_(i2). The first party transmits the value x_(i2) to the second party and this value is the share of x_(i) that is received and held by the second party ([x_(i2)]). Since the value x_(i2) is merely a random value that is unrelated to the original value x_(i), this random value may be transmitted through a data network and be observed by all parties in the group and adversaries without compromising the privacy of x_(i). The share distribution operation can be extended to three or more parties by the generation of additional random values, one for each additional party, and the subtraction of all the random values from the original value by the first party to produce a share for each party. As described above, the value x need not be encrypted to perform the share distribution operation since each of the other parties only receives a random value, although the share distribution operation can also be applied to an encrypted value if x is encrypted. Both of the resharing and the share distribution operations produce shares of a value that are held by multiple parties. The resharing operation additionally prevents a single party from having information about the shares held by other parties, while the share distribution operation enables the first party that holds the original plaintext value to have information about all of the shares, but the other parties do not to observe the share of the first party.

Previously known inner product operations using SPDZ would require executing protocol ReShare 2N times, protocol Mult N times and protocol Add one time. Moreover, previously known inner product operations consume N triples generated in the offline phase, because the number of triples is linearly related to the number of multiplications (Mult) needed.

This disclosure proposes an optimized approach to secure inner product computations which can reduce the number of multiplications needed and therefore decrease the number of triples used. The basic idea of the optimization method is based on the following observation:

(w₁ + x₂)(w₂ + x₁) − x₁x₂ − w₁w₂ = w₁w₂ + w₁x₁ + w₂x₂ +   x₁x₂ −   x₁x₂ − w₁w₂ = x₁w₁ + x₂w₂

If [w₁], [x₁], [x₂], [x₁x₂] and [w₁w₂] can be well established, then only one multiplication is needed (i.e., multiplication of w₁+x₂ and w₂+x₁. Fortunately, since Party P₁ holds x₁ and x₂, then [x₁x₂] can be distributed in a way similar to that of x₁ or x₂. Moreover, since Party P₂ holds w₁ and w₂, then [w₁w₂] can be distributed in a way similar to that of w₁ or w₂.

Based on the above observation, an optimized secure inner product computation protocol (Protocol IP⁺) can be provided. The protocol incurs N/2 multiplication operations in the online phase for the inner product computation, and therefore reduces the number of triples to N/2 as well.

Protocol IP⁺([P₁, x₁, . . . , x_(N)], [P₂, w₁, . . . w_(N)]) where party P₁ holds a private vector x={x₁, . . . , x_(N)}, and party P₂ holds a private vector w={w₁, w_(N)}. N=2k is assumed for simplicity.

-   -   1: For i=1, . . . , N, Party P₁ chooses a value x_(i2) uniform         at random, sets x_(i)=x_(i)−x_(i2), generates ct_(λx) _(i)         =Enc_(pk)(x_(i))         ct_(λ), sends x_(i2) to Party 2 and runs the protocol         ReShare(ct_(λx), P₁, P₂), so that x_(i) is securely shared         between party P₁ and party P₂, denoted by [x_(i)].     -   2: For i=1, . . . , k, Party P₁ computes x_(l) =x_(2i−1)x_(2i),         chooses a value x_(i2) uniform at random, sets x_(i1) =x_(l)         −x_(i2) , generates

${ct}_{\lambda \; {\overset{\_}{x}}_{\iota}} = {{Enc}_{pk}\left( \overset{\_}{x_{\iota}} \right)}$

z,44 ct_(λ), sends x_(i2) to Party 2 and runs the protocol ReShare(ct_(λ) _(x) , P₁, P₂), so that x_(l) is securely shared between party P₁ and party P₂, denoted by [x_(i) ].

-   -   3: For i=1, . . . , N, Party P₂ chooses a value withd it uniform         at random, sets w_(i1)=w_(i)−w_(i2), generates ct_(λw) _(i)         =Enc_(pk)(w_(i))         ct_(λ), sends w_(i1) to Party 1 and runs the protocol         ReShare(ct_(λx), P₁, P₂), so that x_(i) is securely shared         between party P₁ and party P₂, denoted by [w_(i)].     -   4: For i=1, . . . , k, Party P₂ computes w_(l) =w_(2i−1)w_(2i),         chooses a value w_(i1) uniform at random, sets w_(i1) =w_(i)         −w_(i2) , generates

${ct}_{\lambda \; {\overset{\_}{x}}_{\iota}} = {{Enc}_{pk}\left( \overset{\_}{w_{\iota}} \right)}$

ct_(λ), sends w_(i1) to Party 1 and runs the protocol ReShare(ct_(λ) _(w) , P₁, P₂), so that w_(l) is securely shared between party P₁ and party P₂, denoted by [w_(l) ].

-   -   5: For i=1, . . . , k, Parties P₁ and P₂ run SPDZ protocol         Add([w_(2i−1)], [x_(2i)]) and Add([w_(2i)], [x_(2i−1)]), which         results in [w_(2i−1)+x_(2i)] and [w_(2i)+x_(2i−1)], the shartes         of (w_(2i−1)+x_(2i)) and (w_(2i)+x_(2i−1)) respectively. Parties         P₁ and P₂ then execute the protocol Mult([w_(2i−1)+x_(2i)],         [w_(2i)+x_(2i−1)]), the result of which is denoted as [d_(i)].     -   6: Parties P₁ and P₂ run the protocol

Add([d₁], . . . , [d_(l)], −[x₁ ], . . . , −[x_(k) ], −[w₁ ], . . . , −[w_(k) ])

to determine the inner product (d).

We observe that IP⁺ reduces the number of necessary executions of protocol Mult from N to N/2 at the cost of sharing N more private values. However, the cost of executing protocol Add can be ignored because it is ran locally and can be neglected compared to the protocol Mult). However, note that sharing one private value only requires one execution of protocol Reshare, but generating one triple requires 4 executions of protocol ReShare. Moreover, it is worth noting that executing Mult demands interactions between parties (in order to reveal the intermediate values ∈ and ρ shown in protocol Mult), reducing the number of executions of Mult from N to N/2 also reduces the communication and computational complexity. Since the computational complexity of the protocol Add is linear to the input, IP requires 2N−2 arithmetic additions while IP⁺ incurs 5N−2 arithmetic additions.

The remainder of the disclosure will be directed towards background information pertaining to the SPDZ protocol and other pertinent protocols which are used with or may be used with the method described above.

The secret sharing scheme in protocol SPDZ is the n out of n additive secret sharing scheme. According to the additive secret sharing scheme, given a secret value α and a set of parties {P₁, . . . , P_(n)}, α₁, . . . , α_(n−1) uniform is chosen at random and α_(n) is selected such that it satisfies α₁+ . . . +α_(n)=α, where α_(i) is distributed to the party P_(i). As a result, n parties (i.e., all parties) are required in order to recover the secret value α.

In order to tolerate at most n−t unavailable parties, t available parties should be able to recover the secret shares held by those n−t unavailable parties. Based on this observation, protocol ReShare is proposed, where each party redistributes its holding secret share to any subset of t parties, and the protocol ReCombine, where t parties can recover the secret shares of the unavailable parties, and then reconstruct the secret value eventually if needed. Based on the two protocols, a secret share redistribution phase and a secret share recombination phase are proposed which can be placed between the offline and online phases of Protocol SPDZ, so that the resulting SPDZ protocol enjoys the fault tolerance property.

The protocol ReShare is depicted in FIG. 5. The first loop redistributes the holding secret shares of each party to all available parties via authenticated confidential channels. The second loop sums the redistributed secret shares and stores the sums in association with the unique identifiers assigned to each party for each party. The aggregate operation here reduces the storage overhead. Note that the label string “Label” is used to uniquely identify the parties in the subset and the associated redistributed secret shares, and can facilitate reconstructing secret shares. It is worth noting that any secret share is redistributed to the subsets of t parties in the additive manner, therefore any t−1 parties (except the party holding that secret share) cannot recover the secret share. Given n parties, the number of the subsets having t parties is

$\begin{pmatrix} n \\ t \end{pmatrix}{\quad;}$

therefore, the number of subsets including one specific party is

${\begin{pmatrix} n \\ t \end{pmatrix}/n} = {\begin{pmatrix} {n - 1} \\ t \end{pmatrix}.}$

Hence, each party needs to store

$\begin{pmatrix} {n - 1} \\ t \end{pmatrix}\quad$

reaistnoutea secret snares and the respective labels.

Protocol ReCombine is depicted in FIG. 6. Protocol ReCombine enables t parties to use their holding secret shares and redistributed secret shares, to compute secret shares of the secret value. Each party P_(i), 1≤i≤n holds the secret share α_(i) and the set of redistributed secret shares AggSet_(i).

An example will now be discussed to illustrate the protocol ReShare and ReCombine. Suppose there are four parties (i.e., n=4), holding the secret shares x₁, x₂, x₃ and x₄, respectively, such that x=x₁+x₂+x₃+x₄ where x is the secret value. Suppose the system requires tolerating at most two unavailable parties (i.e., n−t=2) meanings that any t=2 parties should be able to reconstruct the secret value.

FIG. 7 depicts a table that presents the intermediate results after party P₁ redistributes its holding secret share x₁ to other parties. Note that “Label” is the string concatenating all parties' indexes in order, and it can uniquely identify each subset of t parties. In the example, since t=2, the set of all subsets having 2 parties is {{P₁, P₃}, {P₂, P₄}, {P₃, P₄}}. The sum of redistributed shares in the same row is equal to the secret share x₁. For example, x_(1,2,Label) _(2,3) +x_(1,3,Label) _(d,3) =x_(1,2,Label) _(2,3) +x_(1,4,Label) _(2,4) =x₁ and x_(1,3,Label) _(3,4) +x_(1,4,Label) _(3,4) =x₁. FIG. 8 shows the final result after all parties redistribute their holding secret shares to the other parties.

FIG. 9 depicts the final result after executing the protocol ReShare. According to FIG. 9, each party holds three redistributed secret shares and the respective labels, as well as the holding secret share. Accordingly, any two parties in this case can construct the secret value. Suppose that parties P₁ and P₂ are online but parties P₃ and P₄ are unavailable. With the protocol ReCombine, parties P₁ and P₂ compute x₁+x_(x,1,Label) _(1,2) +x_(4,1,Label) _(1,2) and x₂+x_(3,2,Label) _(1,2) +x_(4,2,Label) _(1,2) , respecitvely. We can see that these two locally computed values can recover the secret value x, because x_(3,1,Label) _(1,2) +x_(x,2,Label) _(1,2) =x₃, x_(4,1,Labe3l) _(1,2) +x_(4,2,Label) _(1,2) =x₄ and x₁+x₂+x₃+x₄=x.

This disclosure proposes a secret share redistribution phase based on Protocol ReShare and a secret recombination phase based on Protocol ReCombine. The purpose of the former phase is to let all parties redistribute their holding secret shares to other parties, and the purpose of the latter phase is to allow the parties in the subsect parties to recombine the secret shares, which can be used in the online phase.

After executing the offline phase, two different kinds of data shares need to be redistributed: cryptographic key shares (i.e., private key shares and MAC key shares), and the tuple shares. For the cryptographic key shares, only the MAC key shares are considered because private key shares will not be used in the online phase. Note that for each tuple share, each element within the tuple held by the parties is in the form [x], consisting of the data secret share and MAC value share. Therefore, we only need to apply the protocol ReShare on the data secret share and MAC value share, respectively. FIG. 10 describes the secret share redistribution phase in which the MAC key shares and tuple shares which are generated in the offline phase are redistributed.

During the secret share recombination phase, shown in FIG. 11, Protocol ReCombine is applied to compute the MAC key share and tuple shares. Note that it is possible that there are more than t parties available, but only t parties need to be selected.

As depicted in FIG. 12, given the above two phases, the fault tolerant SPDZ can be built by incorporating the secret share redistribution phase and the secret share recombination phase discussed above between the offline and online phases. Note that before executing the secret share recombination phase, t parties need to be selected. Therefore, t parties need to be available.

Given the parameters n (i.e., the total number of parties) and t (i.e., the least number of available parties), the following metrics are considered to compare the SPDZ protocol with the fault tolerant SPDZ protocol:

-   -   Storage consumption: the storage consumption each party spends         on storing a secret value.     -   Number of malicious parties: the number of malicious parties         that are allowed to be present in order to assure the security         of the protocol.     -   Fault tolerance: the number of available parties that are         allowed to be present in order to assure the correctness of the         protocol.

FIG. 13 shows the comparison between the two protocols. As can be seen, the fault tolerant feature for the Fault Tolerant SPDZ (i.e., n−t vs. 0 for SPDZ) is achieved at the price of increasing the storage consumption

$\left( {{i.e.},{\begin{pmatrix} {n - 1} \\ t \end{pmatrix} + {1\mspace{14mu} {{vs}.\mspace{14mu} n}} - 1}} \right.$

for SPDZ) and decreasing the number of malicious parties allowed in the protocol (i.e., t −1 vs. n−1 for SPDZ).

In the above section, it is assumed that participating parties not only perform the offline and online phases, but also provide their private inputs to the computation. The fault tolerant SPDZ scheme described above may also be implemented as a service where multiple service providers host many dedicated servers that are responsible for performing the offline and online phases, and end users upload their private inputs in the form of secret shares ad store them in multiple dedicated servers. Note that the stored input secret shares in the dedicated servers are also in the form of [x]={(x₁, γ₁(x)), . . . , (x, γ_(n), (x))}, where party P₁, 1≤i≤n, holds the tuple (x₁, γ₁(x)), such that x=x₁+ . . . +x_(n), and λx=γ₁(x)+ . . . +γ_(n), (x), and λ is the MAC key. Therefore, it s still valid to apply protocol ReShare and ReCombine on private input shares in the secret share redistribution and secret share recombination phases, so that the extended SPDZ protocol enjoys the fault tolerance under this model.

While the disclosure has been illustrated and described in detail in the drawings and foregoing description, the same should be considered as illustrative and not restrictive in character. It is understood that only the preferred embodiments have been presented and that all changes, modifications and further applications that come within the spirit of the disclosure are desired to be protected. 

what is claimed is:
 1. A method for implementing a secure multiparty inner product computation protocol wherein a first computational party holds a first private vector (x={x₁, . . ., x_(N)}) and a second computational party holds a second private vector (w={w₁, . . . , w_(N)}), the first private vector and the second private vector each having a number N of values, wherein the first computational party and the second computational party want to jointly compute an inner product (d) of the first private vector and the second private vector such that x₁w₁+ . . . +x_(N)w_(N)=x_(i)w_(i), without the first computational party learning the second private vector and the second computational party learning the first private vector, the method comprising: using an SPDZ protocol to securely share the first private vector having a from the first computational party to the second computational party; computing, for i=1, . . . , k, a vector x_(l) ={x₁ , . . . , x_(k) } for the first computational party, where k=N/2 and the value x_(l) =x_(2i−1)x_(2i), and using the SPDZ protocol to securely share the vector x_(l) with the second computational party; using the SPDZ protocol to securely share the second private vector from the first computational party to the second computational party; computing, for i=1, . . . , k, a vector w_(l) ={w₁ , . . . , w_(k) } for the second computational party, where the value w_(l) =w_(2i−1)w_(2i), and using the SPDZ protocol to securely share the vector W, with the first computational party; computing, for i=1, . . . , k, shares [w_(2i−1)+x_(2i)] and [w_(2i) 30 x_(2i−1)] by performing SPDZ protocol Add([w_(2i)], [x_(2i−1)]) and Add([w_(2i)], [x_(2i−1)]), respectively, with the first computational party and the second computational party, the shares [w_(2i−1)+x_(2i)] and [w_(2i)+x_(2i−1)] being shares of (w_(2i−1)+x_(2i)) and (w_(2i)+x_(2i−1)), respectively; computing, for i=1, . . . , k, inner product shares [d_(i)] by performing SPDZ protocol Mult([w_(2i−1)+x_(2i)], [w_(2i)+x_(2i−1)]) with the first computational party and the second computational party; and computing the inner product (d) by performing SPDZ protocol Add([d₁], . . . , [d_(k)], −[x₁ ], . . . , −[x_(k) ], −[w₁ ], . . . , −[w_(k) ]) with the first computational party and the second computational party.
 2. The method of claim 1, wherein the first private vector is data owned by the first computational party and the second private vector is a weight vector.
 3. The method of claim 2, wherein the second computational party is a machine learning service provider.
 4. The method of claim 1, wherein the SPDZ sharing protocol is an additive secret sharing protocol.
 5. The method of claim 4, wherein the first computational party has a first message authentication code (MAC) key share of a MAC key and the second computational party has a second MAC key share of the MAC key such that the first MAC key share and the second MAC key share added together results in the MAC key.
 6. The method of claim 5, wherein the first MAC key is shared by the first computational party to the second computational party and the second MAC key is shared by the second computational party to the first computational party as part of the SPDZ sharing protocol.
 7. The method of claim 1, wherein computing the shares [w_(2i−1)+x_(2i)] and [w_(2i)+x_(2i−1)], inner product shares [d_(i)], and the inner product (d) occurs during an online phase of the SPDZ protocol.
 8. The method of claim 7, wherein both the first computational party and the second computational party generate Beaver triples during an offline phase of the SPDZ protocol which are used by the first computational party and the second computational party during the online phase.
 9. A multiparty computation system comprising: a first computational party having a first private vector (x={x₁, . . . , x_(N)}), where N is a predetermined number; a second computational party having a second vector (w={w₁, . . . , w_(N)}); and at least one secure communication channel via which the first computational party and the second computational party can securely transmit messages to each other; wherein the first computational party and the second computational party are configured to perform a secure multiparty inner product computation based on an SPDZ protocol. to jointly compute an inner product (d) of the first private vector and the second private vector such that x₁w₁+ . . . +x_(N)w_(N)=Σ_(i=1) ^(N)x_(i)w_(i), without the first computational party learning the second private vector and the second computational party learning the first private vector, wherein the secure multiparty inner product computation includes: using the SPDZ protocol to securely share the first private vector having a from the first computational party to the second computational party, computing, for i=1, . . . , k, a vector x_(l) ={x₁ , . . . , x_(k) } for the first computational party, where k=N/2 and the value x_(l) =x_(2i−1)x_(2i), and using the SPDZ protocol to securely share the vector x_(l) with the second computational party, using the SPDZ protocol to securely share the second private vector from the first computational party to the second computational party, computing, for i=1, . . . , k, a vector w_(l) ={w₁ , . . . , w_(k) } for the second computational party, where the value W, =w_(2i−1)w_(2i), and using the SPDZ protocol to securely share the vector W, with the first computational party, computing, for i=1, . . ., k, shares [w_(2i−1)+x_(2i)] and [w_(2i)+x_(2i−1)] by performing SPDZ protocol Add([w_(2i)], [x_(2i−1)]) and Add([w_(2i)], [x_(2i−1)]), respectively, with the first computational party and the second computational party, the shares [w_(2i−1)+x_(2i)] and [w_(2i)+x_(2i−1)] being shares of (w_(2i−1)+x_(2i)) and (w_(2i)+x_(2i−1)), respectively, computing, for i=1, . . . , k, inner product shares [d_(i)] by performing SPDZ protocol Mult([w_(2i−1)+x_(2i)], [w_(2i)+x_(2i−1)]) with the first computational party and the second computational party, and computing the inner product (d) by performing SPDZ protocol Add([d₁], . . . , [d_(k)], −[x₁ ], . . . , −[x_(k) ], −[w₁ ], . . . , −[w_(k) ]) with the first computational party and the second computational party.
 10. The system of claim 9, wherein the first private vector is data owned by the first computational party and the second private vector is a weight vector.
 11. The system of claim 10, wherein the second computational party is a machine learning service provider.
 12. The system of claim 9, wherein the SPDZ sharing protocol is an additive secret sharing protocol.
 13. The system of claim 12, wherein the first computational party has a first message authentication code (MAC) key share of a MAC key and the second computational party has a second MAC key share of the MAC key such that the first MAC key share and the second MAC key share added together results in the MAC key.
 14. The system of claim 13, wherein the first MAC key is shared by the first computational party to the second computational party and the second MAC key is shared by the second computational party to the first computational party as part of the SPDZ sharing protocol.
 15. The system of claim 9, wherein computing the shares [w_(2i−1)+x_(2i)] and [w_(2i)+x_(2i−1)], inner product shares [d_(i)], and the inner product (d) occurs during an online phase of the SPDZ protocol.
 16. The system of claim 15, wherein both the first computational party and the second computational party generate Beaver triples during an offline phase of the SPDZ protocol which are used by the first computational party and the second computational party during the online phase. 